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Solved Problems-18

Problems-18
In the network of the figure, the switch S has been closed for a long time. The switch is suddenly opened at t = 0 and reclosed at t = 20μs. Find expression for voltage V0 for t ≤ 20 μs and t > 20 μs.
 
Description: Description: 4168.png
 
Solution
With the switch closed, the initial voltage across the capacitor is,
Description: Description: 9056.png
 
Description: Description: 4182.png
 
After the switch is opened, the transformed circuit is shown in the figure below.
 
Description: Description: 4189.png
 
Applying KCL at node X, we get,
Description: Description: 9062.png
 
⇒ Description: Description: 9071.png
 
⇒ Description: Description: 9077.png
 
Description: Description: 9094.pngDescription: Description: 9101.png
 
Taking inverse Laplace transform,
Description: Description: 9107.png 
 
Therefore, the desired voltage is,
Description: Description: 9113.png for 0 ≤ t ≤ 20 μs
 
At t = 20 μs, the voltage of node X is,
Description: Description: 9119.png
 
When the switch is reclosed at t = 20 μs, the voltage across the capacitor will be 48.925V. After reclosing the switch, the transformed circuit is shown in the figure below.
 
Description: Description: Description: 1674.png
 
Now let the voltage of node X be VX’. Applying KCL at node X, we get,
Description: Description: 9125.png
⇒ Description: Description: 9133.png
⇒ Description: Description: 9143.png
⇒ Description: Description: 9149.png
⇒ Description: Description: 9158.png
 
Taking inverse Laplace transform, we get,
Description: Description: 9164.png
 
In this case, the output voltage V0 is equal to VX’(t). Since time t is to be counted from the instant the switch is reclosed, t is replaced by Description: Description: 9170.png.
 
∴ Description: Description: 9176.png for t > 20 μs
 




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